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Institute for Plasma Focus Studies

Internet Workshop on Numerical Plasma Focus Experiments

(Supplementary Notes for Sessions 7-8)


[in part extracted from file 2Theory.pdf  from:  http://www.intimal.edu.my/school/fas/UFLF/ ] and from various papers



Radiation Terms


The Bremsstrahlung loss term may be written as:

Recombination loss term is written as:


The line loss term is written as:



where dQ/dt is the total power gain/loss

of the plasma column

By this coupling, if, for example, the radiation loss  is severe, this would lead to a large value of   inwards.  In the extreme case, this leads to radiation collapse, with rp going rapidly to zero, or to such small values that the plasma becomes opaque to the outgoing radiation, thus stopping the radiation loss. 


This radiation collapse occurs at a critical current of 1.6 MA (the Pease-Braginski current) for deuterium.  For gases such as Neon or Argon, because of intense line radiation, the critical current is reduced to even below 100kA, depending on the plasma temperature.


Plasma Self Absorption and transition from volumetric emission to surface emission


Plasma self absorption and volumetric (emission described above) to surface emission of the pinch column have been implemented in the following manner.


The photonic excitation number (see File 3 Appendix by N A D Khattak) is written as follows:

M = 1.66 x 10 -15rp Zn 0.5 ni / (Z T1.5)  with T in eV, rest in SI units

The volumetric plasma self-absorption correction factor A is obtained in the following manner:

A1 = (1 + 10-14ni Z) / (T 3.5))

A2 = 1 / AB1

A = A2 (1 + M)


Transition from volumetric to surface emission occurs when the absorption correction factor goes from 1 (no absorption) down to 1/e (e=2.718) when the emission becomes surface-like given by the expression:

where the constant const is taken as 4.62x10-16 to conform with numerical experimental observations that this value enables the smoothest transition, in general, in terms of power values from volumetric to surface emission.


Where necessary another fine adjustment is made at the transition point adjusting the constant so that the surface emission power becomes the same value as the absorption corrected volumetric emission power at the transition point. Beyond the transition point (with A less than 1/e) radiation emission power is taken to be the surface emission power.


Neutron Yield



Adapted from the following papers (with modifications for erratum)

Pinch current limitation effect in plasma focus (This version includes an Erratum)

S. Lee and S. H. Saw, Appl. Phys. Lett. 92, 021503 (2008), DOI:10.1063/1.2827579

Copyright (2008) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. This article appeared in (citation above) and may be found at



Neutron Scaling Laws from Numerical Experiments (This version includes an Erratum)

S Lee and S H Saw, J of Fusion Energy, DOI: 10.1007/s10894-008-9132-7

published first online 20 February 2008 at http://dx.doi.org/10.1007/s10894-008-9132-7  

"The original publication is available at www.springerlink.com."


Neutron yield is calculated with two components, thermonuclear term and beam-target term.

The thermonuclear term is taken as:

dYth = 0.5ni2( 3.142)rp2zf<sv>(time interval)

Where <sv> is the thermalised fusion cross section-velocity product corresponding to the plasma temperature, for the time interval under consideration. The yield Yth is obtained by summing up over all intervals during the focus pinch.


The beam-target term is derived using the following phenomenological beam-target neutron generating mechanism17, incorporated in the present RADPFV5.13. A beam of fast deuteron ions is produced by diode action in a thin layer close to the anode, with plasma disruptions generating the necessary high voltages. The beam interacts with the hot dense plasma of the focus pinch column to produce the fusion neutrons. In this modeling each factor contributing to the yield is estimated as a proportional quantity and the yield is obtained as an expression with proportionality constant. The yield is then calibrated against a known experimental point.

          The beam-target yield is written in the form:   Yb-t ~nb ni (rp2zp) (s vb) t

where nb is the number of beam ions per unit plasma volume, ni is the ion density, rp is the radius of the plasma pinch with length zp, s the cross-section of the D-D fusion reaction, n- branch18, vb the beam ion speed and t is the beam-target interaction time assumed proportional to the confinement time of the plasma column.

          Total beam energy is estimated17 as proportional to LpIpinch2, a measure of the pinch inductance energy, Lp being the focus pinch inductance. Thus the number of beam ions is Nb~LpIpinch2/vb2 and nb is Nb divided by the focus pinch volume. Note that Lp~ln(b/rp)zp , that4 t~rp~zp , and that vb~U1/2 where U is the disruption-caused diode voltage17. Here ‘b’ is the cathode radius. We also assume reasonably that U is proportional to Vmax, the maximum voltage induced by the current sheet collapsing radially towards the axis.


          Hence we derive: Yb-t= Cn ni Ipinch2zp2((lnb/rp))s/Vmax1/2                                (1)


 where Ipinch is the current flowing through the pinch at start of the slow compression phase; rp and zp are the pinch dimensions at end of that phase. Here Cn is a constant which in practice we will calibrate with an experimental point.

          The D-D cross-section is highly sensitive to the beam energy so it is necessary to use the appropriate range of beam energy to compute s.  The code computes Vmax of the order of 20-50 kV. However it is known17, from experiments that the ion energy responsible for the beam-target neutrons is in the range 50-150keV17, and for smaller lower-voltage machines the relevant energy 19 could be lower at 30-60keV. Thus to align with experimental observations the D-D cross section s is reasonably obtained by using beam energy equal to 3 times Vmax.

          A plot of experimentally measured neutron yield Yn vs Ipinch was made combining all available experimental data2,4,12,13,17,19-22. This gave a fit of Yn=9x1010Ipinch3.8 for Ipinch in the range 0.1-1MA. From this plot a calibration point was chosen at 0.5MA, Yn=7x109 neutrons. The model code23 RADPFV5.13 was thus calibrated to compute Yb-t which in our model is the same as Yn.



Notes on The total current and Ipeak, the plasma current and Ipinch

Extracted From: Computing Plasma Focus Pinch Current from Total Current Measurement     

S. Lee, S. H. Saw, P. C. K.  Lee, R. S. Rawat and H. Schmidt, Appl Phys Letters 92, 111501 (2008) DOI:10.1063/1.2899632


The total current Itotal waveform in a plasma focus discharge is easily measured using a Rogowski coil. The peak value Ipeak of this trace is commonly taken as a measure of the drive efficacy and is often used to scale the yield performance of the plasma focus. This is despite the fact that yields should more consistently be scaled to focus pinch current Ipinch, since it is Ipinch which directly powers the emission processes. The reason many researchers use Ipeak instead of Ipinch for scaling is simply that while Ipeak is easily measured, Ipinch, which is the value of the plasma sheath current Ip at time of pinch, is very difficult to measure even in large devices where it is possible to place magnetic probes near the pinch. This measurement is also inaccurate and perturbs the pinch. In a small device, there is no space for such a measurement.


The relationship between Ipinch and Ipeak is not simple and has only been recently elaborated. It primarily depends on the value of the static inductance L0 compared to the dynamic inductances of the plasma focus. As L0 is reduced, the ratio Ipinch / Ipeak drops. Thus, yield laws scaled to Ipeak will not consistently apply when comparing two devices with all parameters equal but differing significantly in L0. Better consistency is achieved when yield laws are scaled to Ipinch. In this paper, we propose a numerical method to consistently


Distinguishing the Itotal waveform from the Ip waveform


A measured trace of Itotal is commonly obtained with a Rogowski coil wrapped around the plasma focus flange through which is fed Itotal discharged from the capacitor bank between the coaxial electrodes across the back wall. A part of Itotal, being the plasma sheath current Ip, lifts off the back-wall insulator and drives a shock wave axially down the coaxial space. We denote fc as the current fraction Ip/Itotal for the axial phase and fcr for the radial phases. In modeling it is found that a reasonable value for initial trial for fc is 0.7 with a similar first trial value for fcr . However in a DPF78 experiment fc was found to vary from 0 at the start of the axial phase rising rapidly above 0.6 for the rest of the axial phase. In the radial phase fcr was found to stay above 0.6 before dropping to 0.48 at the start of the pinch and then towards 0.4 as the pinch phase progressed. These Stuttgart results confirm a complex relationship between the waveforms of Itotal and Ip.


The performance of a plasma focus is closely linked to the current Ipinch actually participating in the focus pinch phase rather than the total current flowing in the circuit. It is a common practice to take Ipeak or some representative fraction of it as Ipinch. Another practice is to take the value of Itotal at the time of the pinch as Ipinch Whilst in their special cases this practice could be justifiable, the distinction of Ip from Itotal should generally be clearly made. We emphasize that it should be the value of Ip at the time of pinch which is the relevant value for the purpose of yield scaling. The practice of associating yield scaling with the total current waveform (i.e. taking Ipeak or Itotal at estimated pinch time) would be justifiable if there were a linear relationship between the waveforms of Itotal and Ip. However as shown by the Stuttgart experiments the actual relationship is a very complex one which we may ascribe to the interplay of the various electro-dynamical processes including the relative values of static inductance Lo, tube inductance and the dynamic resistances which depend on the tube geometry and plasma sheath speeds. This relationship may change from one machine to the next. Whilst these electro-dynamical processes and other relevant ones such as radiation are amenable to modeling there are other machine effects such as back wall restriking (for example due to high induced voltages during the pinch phase) which can almost unpredictably affect the relationship between Itotal and Ip during the crucial radial phases. Hence it is not only simplistic to discuss scaling in terms of the Itotal waveform (i.e. taking Ipeak or the value of Itotal at the estimated time of pinch) but also inconsistent. One of the most important features of a plasma focus is its neutron production. The well-known neutron yield scaling, with respect to current, based on various compilations of experimental data, is Yn Ipinchx where x is varied in the range 3–5. In a recent paper , numerical experiments using a code was used to derive a scaling with x = 4.7. Difficulties in the interpretation of experimental data ranging across big and small plasma focus devices include the assignment of the representative neutron yield Yn for any specific machine and the assignment of the value of Ipinch. In a few larger machines attempts were made to measure Ipinch using magnetic probes placed near the pinch region, with uncertainties of 20%. Moreover the probes would have affected the pinching processes. In most other cases related to yield scaling data compilation or interpretation Ipinch is simply assigned a value based on the measurement of peak total current Ipeak or the value of total current at the observed current dip.


The difficulties in distinguishing Ipinch from Itotal are obviated in numerical experiments using the Lee Model [In a typical simulation, the Itotal trace is computed and fitted to a measured Itotal trace from the particular focus. Three model parameters for fitting are used: axial mass swept-up factor fm, current factor fc and radial mass factor fmr. A fourth model parameter, radial current factor, fcr may also be used. When correctly fitted the computed Itotal trace agrees with the measured I  trace in peak amplitude, rising slope profile and topping profile which characterize the axial phase electro-dynamics. The radial

phase characteristics are reflected in the roll-over of the current trace from the flattened top region, and the subsequent current drop or dip. Any machine effects, such as restrikes, current sheath leakage and consequential incomplete mass swept up, not included in the simulation physics is taken care of by the final choice of the model parameters, which are fine-tuned in the feature-by-feature comparison of the computed Itotal trace with the measured Itotal trace. Then there is confidence that the computed gross dynamics, temperature, density, radiation, plasma sheath currents, pinch current and neutron yield may also be realistically compared with experimental values.


A note on scaling:


Scaling of yields to say Ipinch should be carried out using yields which are at optimum, or at least near optimum. If one indiscriminately uses any data one may end up with completely trivial or misleading results. For example if a point is used at too high or low pressure (away from the optimum pressure) then there may be zero yield ascribed to values of Ipinch.





1 Lee S 1984 Radiations in Plasmas ed B McNamara (World Scientific) pp 978–87

Also: S. Lee in Laser and Plasma Technology, edited by S. Lee, B. C. Tan, C. S.

Wong, & A. C. Chew. World Scientific, Singapore, (1985), pp. 387–420.


2S. Lee, T. Y. Tou, S. P. Moo, M. A. Elissa, A. V. Gholap, K. H. Kwek, S.

Mulyodrono, A. J. Smith, Suryadi, W. Usala, & M. Zakaullah, Am. J.

Phys. 56, 62 (1988).


3T. Y. Tou, S. Lee, & K. H. Kwek, IEEE Trans. Plasma Sci. 17, 311 (1989).


4S. Lee and A. Serban, IEEE Trans. Plasma Sci. 24, 1101 (1996).


5D. E. Potter, Phys. Fluids 14, 1911 (1971).


6M. H. Liu, X. P. Feng, S. V. Springham, and S. Lee, IEEE Trans. Plasma

Sci. 26, 135(1998).


7S. Lee, P. Lee, G. Zhang, X. Feng, V. A. Gribkov, M. Liu, A. Serban, and

T. Wong, IEEE Trans. Plasma Sci. 26, 1119 (1998).


8S. Bing, “Plasma dynamics and x-ray emission of the plasma focus,” Ph.D. thesis, NIE, (2000) in ICTP Open Access Archive: http://eprints.ictp.it/99/


9S. Lee, in http://ckplee.myplace.nie.edu.sg/plasmaphysics/   (2000 & 2007).


10S. Lee in ICTP Open Access Archive:  http://eprints.ictp.it/85/    (2005).


11V. Siahpoush, M. A. Tafreshi, S. Sobhanian, and S. Khorram, Plasma

Phys. Controlled Fusion 47, 1065 (2005).


12S. Lee, Twelve Years of UNU/ICTP PFF-A Review (1998) IC, 98 (231);

A.Salam ICTP, Miramare, Trieste ( in ICTP OAA: http://eprints.ictp.it/31/  ).


13L. Soto, P. Silva, J. Moreno, G. Silvester, M. Zambra, C. Pavez, L. Altamirano, H. Bruzzone, M. Barbaglia, Y. Sidelnikov, and W. Kies, Braz. J. Phys. 34, 1814 (2004).


14H. Acuna, F. Castillo, J. Herrera, and A. Postal, International Conference

on Plasma Sci, 3–5 June 1996 (unpublished), p. 127.


15C. Moreno, V. Raspa, L. Sigaut,& R. Vieytes, Appl. Phys. Lett. 89, 15 (2006).


16D. Wong, P. Lee, T. Zhang, A. Patran, T. L. Tan, R. S. Rawat, and S. Lee,

Plasma Sources Sci. Technol. 16, 116 (2007).


17V A Gribkov, A Banaszak, B Bienkowska, A V Dubrovsky, I Ivanova-Stanik,

L Jakubowski, L Karpinski, R A Miklaszewski, M Paduch, M J Sadowski, M Scholz,

A. Szydlowski, and K. Tomaszewski, J. Phys. D 40, 3592 (2007).


18J. D. Huba, 2006 Plasma Formulary, p. 44. http://wwwppd.nrl.navy.mil/nrlformulary/NRL_FORMULARY_07.pdf


19S. V. Springham, S. Lee, and M. S. Rafique, Plasma Phys. Controlled

Fusion 42, 1023 (2000).


20W. Kies, in Laser and Plasma Technology, Proceedings of Second Tropical

College, edited by S. Lee, B. C. Tan, C. S. Wong, A. C. Chew, K. S. Low, H. Ahmad, and Y. H. Chen World Scientific, Singapore, (1988), pp. 86–137.


21H. Herold, in Laser and Plasma Technology, Proceedings of Third Tropical

College, edited by C. S. Wong, S. Lee, B. C. Tan, A. C. Chew, K. S.

Low, and S. P. Moo World Scientific, Singapore, (1990), pp. 21–45.


22A. Patran, R. S. Rawat, J. M. Koh, S. V. Springham, T. L. Tan, P. Lee, & S. Lee, 31st EPS Conf on Plasma Phys London, (2004), Vol. 286, p. 4.213.


23S Lee  Radiative Dense Plasma Focus Computation Package (2008) RADPF:  http://www.intimal.edu.my/school/fas/UFLF/



Reference to this course and the Lee model code should be given according to the following format:


Lee S.  Radiative Dense Plasma Focus Computation Package (2008) : RADPF              www.plasmafocus.net      www.intimal.edu.my/school/fas/UFLF/